Video surveillance plays an important role in public security. To store the growing volume of surveillance videos, video compression is beneficial for reducing video volume; however, it is simultaneously harmful to the video quality. Video quality assessment (VQA) methods help to achieve a tradeoff between the data volume and perceptual quality of compressed surveillance videos. Generally speaking, surveillance video quality assessment (SVQA) is different from conventional VQA because surveillance videos are usually used for specific tasks, e.g., pedestrian recognition, rather than for entertainment purposes. Therefore, in this work, we propose two full-reference SVQA methods based on the concept of Quality of Recognition (QoR). We first design two new tasks, distorted face verification (DFV) and distorted face identification (DFI), based on which we further propose two SVQA methods, DFV-SVQA and DFI-SVQA, and corresponding quality metrics. The core components of the DFVSVQA and DFI-SVQA methods are feature extractors (a DFV model and a DFI model), which we construct using convolutionalneural- network-based face recognition models. In addition, we construct a real-world surveillance video dataset, based on which we analyze how various factors, including the video codec, compression level, face resolution and light intensity, affect the quality of compressed surveillance videos. We find that compared with conventional VQA methods, our methods are more effective in measuring the quality of surveillance videos while maintaining an acceptable time efficiency.